Elastography imaging offers great opportunities for noninvasive medical diagnostics. It provides quantitative, real-time estimation of tissue elasticity similar to manual palpation, but with a much higher degree of resolution. The advantage of this technique is the ability to evaluate stiffness of internal tissues and organs [Ophir et al. 1996]. Since the shear moduli of tissues vary over wide ranges, elastography is capable of high image contrast as compared to medical ultrasound. For example, in normal and pathological soft tissues, shear moduli can differ from each other by up to 8 times whereas the compressional moduli are nearly identical. Recent investigation of nonlinear elastic properties has shown that nonlinear parameters are more sensitive to changes in tissue stiffness than linear shear moduli [Zheng et al. 2003]. Nonlinear effects in shear waves in soft elastic media have been measured in Fink's Group [Catheline et al. 2003]. A unique apparatus developed in France allows researchers in this group to observe, in real time, shear wave propagation in tissue and to measure the wave displacements throughout a two-dimensional (2D) plane. In related experiments where a directional shear wave beam has propagated in a tissue phantom, substantial wave distortion has been observed, which suggests the importance of nonlinear elasticity and may provide a basis for imaging the nonlinear elastic properties of tissue [Catheline et. al. 2003]. The primary objective of the project proposed here is to develop a mathematical model based on the investigators' new constitutive equation, which allows 1 to analyze linear and nonlinear propagation of shear wave beams and scattering in soft tissues. Our hypothesis is that an improved shear wave model that includes nonlinearity and diffraction will provide the much needed mathematical basis for generating shear wave elastograms. The resulting elastograms could help to determine quantitatively elastic properties in the form of an image with great contrast between different tissue types. The developed model for nonlinear shear wave beams in tissues will be applied to make accurate, quantitative elastic characterization of a variety of tissues including anisotropic ones. In particular, we are going to analyze and interpret data provided by our French colleagues (Fink's group). Their experimental data will be used to test our model and quantify its accuracy in determination of shear moduli. The long-term goal of the project is to provide the mathematical foundation necessary to implement clinical elastography imaging. Public health implications of this research: Elastography imaging shows promise for earlier detection of tumors than is now possible. Earlier detection offers the possibility of more effective treatment, resulting in the saving of both patients' lives and strained public health resources. In addition, elastography is considerably less expensive than magnetic resonance imaging, affording a further savings of public health funds. [unreadable] [unreadable]